The shape of an index function.

Indices passed to an LMAD. Must always match the rank of the LMAD.

LMAD's representation consists of a general offset and for each dimension a stride, number of elements (or shape), and permutation. Note that the permutation is not strictly necessary in that the permutation can be performed directly on LMAD dimensions, but then it is difficult to extract the permutation back from an LMAD.

LMAD algebra is closed under composition w.r.t. operators such as permute, index and slice. However, other operations, such as reshape, cannot always be represented inside the LMAD algebra.

It follows that the general representation of an index function is a list of LMADS, in which each following LMAD in the list implicitly corresponds to an irregular reshaping operation.

However, we expect that the common case is when the index function is one LMAD -- we call this the "nice" representation.

Finally, the list of LMADs is kept in an LMAD together with the shape of the original array, and a bit to indicate whether the index function is contiguous, i.e., if we instantiate all the points of the current index function, do we get a contiguous memory interval?

By definition, the LMAD \( \sigma + \{ (n_1, s_1), \ldots, (n_k, s_k) \} \), where \(n\) and \(s\) denote the shape and stride of each dimension, denotes the set of points:

\[ \{ ~ \sigma + i_1 * s_1 + \ldots + i_m * s_m ~ | ~ 0 \leq i_1 < n_1, \ldots, 0 \leq i_m < n_m ~ \} \]

A single dimension in an LMAD.

A complete permutation.

Handle the case where a slice can stay within a single LMAD.

Flat-slice an LMAD.

Reshape an LMAD.

There are four conditions that all must hold for the result of a reshape operation to remain in the one-LMAD domain:

1. the permutation of the underlying LMAD must leave unchanged the LMAD dimensions that were *not* reshape coercions.
2. the repetition of dimensions of the underlying LMAD must refer only to the coerced-dimensions of the reshape operation.

If any of these conditions do not hold, then the reshape operation will conservatively add a new LMAD to the list, leading to a representation that provides less opportunities for further analysis

Coerce an index function to look like it has a new shape. Dynamically the shape must be the same.

Permute dimensions.

Shape of an LMAD.

Substitute a name with a PrimExp in an LMAD.

Generalised iota with user-specified offset.

Returns true if two LMADs are equivalent.

Equivalence in this case is matching in offsets and strides.

The largest possible linear address reachable by this LMAD, not counting the offset. If you add one to this number (and multiply it with the element size), you get the amount of bytes you need to allocate for an array with this LMAD (assuming zero offset).

Conceptually expand LMAD to be a particular slice of another by adjusting the offset and strides. Used for memory expansion.

Is this is a row-major array with zero offset?

Returns True if the two LMADs could be proven disjoint.

Uses some best-approximation heuristics to determine disjointness. For two 1-dimensional arrays, we can guarantee whether or not they are disjoint, but as soon as more than one dimension is involved, things get more tricky. Currently, we try to conservativelyFlatten any LMAD with more than one dimension.

Dynamically determine if two LMAD are equal.

True if offset and constituent LMADDim are equal.

Create an LMAD that is existential in everything except shape.

When comparing LMADs as part of the type check in GPUMem, we may run into problems caused by the simplifier. As index functions can be generalized over if-then-else expressions, the simplifier might hoist some of the code from inside the if-then-else (computing the offset of an array, for instance), but now the type checker cannot verify that the generalized index function is valid, because some of the existentials are computed somewhere else. To Work around this, we've had to relax the KernelsMem type-checker a bit, specifically, we've introduced this function to verify whether two index functions are "close enough" that we can assume that they match. We use this instead of `lmad1 == lmad2` and hope that it's good enough.

Turn all the leaves of the LMAD into Exts, except for the shape, which where the leaves are simply made Free.

Retrieve those elements that existentialize changes. That is, everything except the shape (and in the same order as existentialise existentialises them).